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Metric relation describing the position of the incenter

Source: Romanian IMO TST 2006, day 5, problem 4

May 23, 2006
geometryincentercircumcircleparallelogramtrigonometrygeometry proposed

Problem Statement

Let ABCABC be an acute triangle with AB≠ACAB \neq AC. Let DD be the foot of the altitude from AA and ω\omega the circumcircle of the triangle. Let ω1\omega_1 be the circle tangent to ADAD, BDBD and ω\omega. Let ω2\omega_2 be the circle tangent to ADAD, CDCD and ω\omega. Let ℓ\ell be the interior common tangent to both ω1\omega_1 and ω2\omega_2, different from ADAD.
Prove that ℓ\ell passes through the midpoint of BCBC if and only if 2BC=AB+AC2BC = AB + AC.
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